Tools for constructing dynamical models of rhythmic movement

The dynamical approach to movement control, championed by workers at Haskins Laboratories and the University of Connecticut, seeks to describe action phenomena in terms of low-dimensional dynamical equations of motion. A major problem for such an approach is how to construct dynamical models on the basis of empirical data. This problem has a longstanding history in astronomy (Hill, Mathieu, Dulac, Liapunov, Krylov, Poincaré, Picard, since around 1890), electronics and radioengineering (Van der Pol, Andronov, Chaikin, Minorski, Liénard, Corbeiller, since around 1920), wave mechanics (Schrödinger, Mathieu, since around 1930), electromotor theory (Corbeiller, since around 1930), and control theory (Lefschetz, Pontryagin, Pearson, Kalman, since around 1945). In the spirit of these original contributions, a mathematical argument is provided that, for an important class of human movements, only four elementary series of non-linearities are required to describe friction and stiffness. This analysis leads to a catalogue of functions constituting a tool for identifying the essential terms of non-linear differential equations. It also gives rise to new graphical techniques for identifying these non-linear terms and their local prominence. These techniques for constructing testable dynamical models are supplemented and compared with the graphical tools offered by Liénard and the analytical tools offered by Andronov and Pearson. Throughout the paper, the discussed analytical tools will be applied to quantify typical phase-portraits of the hand movement for juggling three balls. The best model of this specific phase-portrait turns out to be one in which discrete, external forcing is taken into account in addition to the autonomous non-linear dynamics of the juggling.